Homotopy-theoretically enriched categories of noncommutative motives

Jack Morava

doi:10.1186/s40687-015-0028-7

Abstract

Waldhausen's $K$-theory of the sphere spectrum (closely related to the algebraic $K$-theory of the integers) is a naturally augmented $S^0$-algebra, and so has a Koszul dual. Classic work of Deligne and Goncharov implies an identification of the rationalization of this (covariant) dual with the Hopf algebra of functions on the motivic group for their category of mixed Tate motives over $\Z$. This paper argues that the rationalizations of categories of non-commutative motives defined recently by Blumberg, Gepner, and Tabuada consequently have natural enrichments, with morphism objects in the derived category of mixed Tate motives over $\Z$. We suggest that homotopic descent theory lifts this structure to define a category of motives defined not over $\Z$ but over the sphere ring-spectrum $S^0$.

Highlights

1.1 Building on earlier work going back at least three decades [26], Deligne and Goncharov have defined a Q-linear Abelian rigid tensor category of mixed Tate motives over the integers of a number field: in particular, the category MTQ(Z) of such motives over the rational integers
We argue here that these objects are analogous to the cells of stable homotopy theory: in that, for example, the image
Deligne and Goncharov’s definition ([27], § 1.6) depends on the validity of the Beilinson-Soulé vanishing conjecture for number fields, which implies that their category MTQ(Z) can be characterized by a very simple spectral sequence with E2-term

Summary

Introduction

1.1 Building on earlier work going back at least three decades [26], Deligne and Goncharov have defined a Q-linear Abelian rigid tensor category of mixed Tate motives over the integers of a number field: in particular, the category MTQ(Z) of such motives over the rational integers. Is a graded-commutative Hopf algebra with one generator in each even degree, canonically isomorphic to the classical algebra of symmetric functions with coproduct x(t) = x(t) ⊗ x(t) (x(t) = x2ktk, x0 := 1). Ii) Hoffman ([40], Theorem 2.5) constructs an isomorphism exp : QSymm∗ ⊗ Q → QSymm∗ ⊗ Q of graded Hopf algebras over the rationals, taking to ; so over Q we can think of the morphism defined by the proposition as the inclusion of the symmetric functions in the quasisymmetric functions with the quasishuffle product. The rational de Rham algebra of forms on a reasonable space is a good model for the rational Spanier-Whitehead commutative ringspectrum [ X, SQ0 ]

Finite-dimensional graded modules over a field k have a good duality functor

The Künneth spectral sequence

Remarks

Example